Journal of Applied Probability

Critical scaling for the SIS stochastic epidemic

R. G. Dolgoarshinnykh and Steven P. Lalley

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We exhibit a scaling law for the critical SIS stochastic epidemic. If at time 0 the population consists of √N infected and N - √N susceptible individuals, then when the time and the number currently infected are both scaled by √N, the resulting process converges, as N → ∞, to a diffusion process related to the Feller diffusion by a change of drift. As a consequence, the rescaled size of the epidemic has a limit law that coincides with that of a first passage time for the standard Ornstein-Uhlenbeck process. These results are the analogs for the SIS epidemic of results of Martin-Löf (1998) and Aldous (1997) for the simple SIR epidemic.

Article information

Source
J. Appl. Probab. Volume 43, Number 3 (2006), 892-898.

Dates
First available in Project Euclid: 20 September 2006

Permanent link to this document
http://projecteuclid.org/euclid.jap/1158784956

Digital Object Identifier
doi:10.1239/jap/1158784956

Mathematical Reviews number (MathSciNet)
MR2274810

Zentralblatt MATH identifier
1119.92056

Subjects
Primary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx]
Secondary: 92D30: Epidemiology

Keywords
Stochastic epidemic model SIS SIR Feller diffusion Ornstein-Uhlenbeck process

Citation

Dolgoarshinnykh, R. G.; Lalley, Steven P. Critical scaling for the SIS stochastic epidemic. J. Appl. Probab. 43 (2006), no. 3, 892--898. doi:10.1239/jap/1158784956. http://projecteuclid.org/euclid.jap/1158784956.


Export citation

References

  • Aldous, D. (1997). Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Prob. 25, 812--854.
  • Darling, D. A. and Siegert, A. J. F. (1953). The first passage problem for a continuous Markov process. Ann. Math. Statist. 24, 624--639.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Characterization and Convergence. John Wiley, New York.
  • Feller, W. (1951). Diffusion processes in genetics. In Proc. 2nd Berkeley Symp. Math. Statist. Prob., 1950, University of California Press, Berkeley and Los Angeles, pp. 227--246.
  • Friedman, A. (1964). Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, NJ.
  • Jiřina, M. (1969). On Feller's branching diffusion processes. Časopis P\v est. Mat. 94, 84--90, 107.
  • Kryscio, R. J. and Lefèvre, C. (1989). On the extinction of the S-I-S stochastic logistic epidemic. J. Appl. Prob. 26, 685--694.
  • Lindvall, T. (1974). On Feller's branching diffusion processes. Adv. Appl. Prob. 6, 309--321.
  • Martin-Löf, A. (1998). The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier. J. Appl. Prob. 35, 671--682.
  • Nasell, I. (1996). The quasi-stationary distribution of the closed endemic SIS model. Adv. Appl. Prob. 28, 895--932.
  • Nasell, I. (1999). On the time to extinction in recurrent epidemics. J. R. Statist. Soc. B 61, 309--330.
  • Norden, R. H. (1982). On the distribution of the time to extinction in the stochastic logistic population model. Adv. Appl. Prob. 14, 687--708.
  • Weiss, G. and Dishon, M. (1971). On the asymptotic behavior of the stochastic and deterministic models of an epidemic. Math. Biosci. 11, 261--265.